CBSE - Mathematics - Arithematic Progressions

For this A.P.,
a = 7
l = 84
d = a₂ − a₁ =- 7 = 21/2 - 7 = 7/2
Let 84 be the n^th term of this A.P.
l = a (n - 1)d
84 = 7 + (n - 1) × 7/2
77 = (n - 1) × 7/2
22 = n − 1
n = 23
We know that,
S_n = n/2 (a + l)
S_n = 23/2 (7 + 84)
= (23×91/2) = 2093/2
= 

(ii) 34 + 32 + 30 + ……….. + 10
For this A.P.,
a = 34
d = a₂ − a₁ = 32 − 34 = −2
l = 10
Let 10 be the n^th term of this A.P.
l = a + (n − 1) d
10 = 34 + (n − 1) (−2)
−24 = (n − 1) (−2)
12 = n − 1
n = 13
S_n = n/2 (a + l)
= 13/2 (34 + 10)
= (13×44/2) = 13 × 22
= 286

(iii) (−5) + (−8) + (−11) + ………… + (−230) For this A.P.,
a = −5
l = −230
d = a₂ − a₁ = (−8) − (−5)
= − 8 + 5 = −3
Let −230 be the n^th term of this A.P.
l = a + (n − 1)d
−230 = − 5 + (n − 1) (−3)
−225 = (n − 1) (−3)
(n − 1) = 75
n = 76
And,
S_n = n/2 (a + l)
= 76/2 [(-5) + (-230)]
= 38(-235)
= -8930